"I will make the man holding the prerogative of the supreme command in the USA pay dearly for his speech calling for totally destroying the DPRK", said the statement carried by North's official Korean Central News Agency on Friday morning.
In his speech, Trump called North Korean regime a "band of criminals" and Kim a "Rocket Man" on "a suicide mission".
US Treasury Secretary Steven Mnuchin said yesterday that firms around the world must now choose between doing business with the United States or North Korea, under broad new sanctions authorised by the United States.
In an official statement released Thursday, Kim called Trump " a frightened dog" and a "gangster fond of playing with fire".
In the margins of the general assembly of the united Nations in NY, the head of the diplomacy of north korea's Ri Yong-ho told journalists that Pyongyang could now be given to detonate a hydrogen bomb out of his territory. Pyongyang already carried out a nuclear test earlier this month, which it claimed to be an H-bomb.
Ministry spokesman Baik Tae-hyun says North Korea should stop provocations that would "lead to its own isolation and demise".
The latest round of rhetoric began on Tuesday when Trump, in his first address to the United Nations, made the threat to destroy North Korea, a country of 26 million people.
Hours later, Kim's foreign minister told reporters in NY that Pyongyang could launch a nuclear missile test in response.
This came just days after Trump threatened to "totally destroy" North Korea if it continued with its nuclear weapons development.
North Korea's foreign minister later said that could be a hydrogen bomb test in the Pacific Ocean if Kim orders it.
Some analysts saw Kim's statement as a clear announcement that North Korea would ramp up its already brisk pace of weapons testing, which has included missiles meant to target US forces throughout Asia and the USA mainland.
The Japanese government on Friday said that it was prepared to face North Korea's threat to launch a hydrogen bomb into the Pacific Ocean. "Perhaps China will put a heavy move on North Korea and end this nonsense once and for all!"
That statement was particularly worrying to Narang, he told CNN. They mark a rare direct intervention by the North Korea leader, who usually lets his deputies issue statements.
"As President Trump said, the global community must continue to work together to confront rogue regimes and that is exactly what the Prime Minister will be doing in discussions with world leaders in NY this week".